For which time intervals, is the percentage rise of population the same for the following data ? Period | Population 1970 | 40,000 1980 | 50,000 1990 | 60,000 2000 | 72,000 2010 | 80,000

Correct — B, (b) 1980 – 90 and 1990 – 2000.

THE STEP THAT DECIDES THIS QUESTION is to compute RATIOS of successive figures rather than DIFFERENCES between them. A percentage rise is the increase divided by the figure it started from, so two intervals can add the same number of people and still show different percentage rises, and two intervals can add different numbers of people and show the same percentage rise. Everything else here is arithmetic.

Take the four decades in turn, using the figures printed in the table inside the stem — 40,000 in 1970, 50,000 in 1980, 60,000 in 1990, 72,000 in 2000 and 80,000 in 2010.

1970 to 1980: 50,000 ÷ 40,000 = 1·25 → a rise of 25 per cent 1980 to 1990: 60,000 ÷ 50,000 = 1·20 → a rise of 20 per cent 1990 to 2000: 72,000 ÷ 60,000 = 1·20 → a rise of 20 per cent 2000 to 2010: 80,000 ÷ 72,000 = 1·111… → a rise of about 11 per cent

Only two of the four match, and they are consecutive: 1980 – 90 and 1990 – 2000, both at 20 per cent. That is option (b).

WHY THE TABLE IS BUILT THE WAY IT IS. Look at the increases rather than the ratios for a moment: +10,000, +10,000, +12,000, +8,000. The first two decades add exactly the same number of people, and that is the whole design of the question. A candidate who scans the column for equal gaps will find 1970 – 80 and 1980 – 90 immediately and stop. But the base has changed underneath him: the first 10,000 is added to 40,000 and the second to 50,000, so the same absolute increase is a smaller proportional one. Meanwhile the pair that does match is the pair whose increases are UNEQUAL — 10,000 and then 12,000 — because the base grew from 50,000 to 60,000 in the same proportion.

That is the general point about growth rates, and it is worth stating as a rule. A series growing at a constant percentage rate is a GEOMETRIC progression, in which each term is a fixed multiple of the one before; a series growing by a constant amount is an ARITHMETIC progression, in which each term is the one before plus a fixed number. The two look similar over a short stretch and diverge sharply over a long one. This table is arithmetic for its first two steps, geometric across its middle two, and then slows down.

A shortcut worth having for the examination hall: to compare two decadal ratios you rarely need to divide. 50 against 40 is one quarter more; 60 against 50 is one fifth more; 72 against 60 is one fifth more again, because 12 is a fifth of 60; 80 against 72 is one ninth more. One quarter, one fifth, one fifth, one ninth — the matching pair falls out without a single long division. All the figures in the column are multiples of 4,000, which is what makes those fractions come out clean.

Why the others are wrong

• (a)1970 – 80 and 1980 – 90 — This is the trap the table was built for, and it is the option a candidate reaches by comparing DIFFERENCES instead of RATIOS. The population rises by exactly 10,000 in 1970 – 80 and by exactly 10,000 again in 1980 – 90, and no other pair of decades in the table shares an absolute increase, so a reader scanning the column for a match finds this one at once. But a percentage rise is measured against the starting figure, and the starting figures differ: 10,000 on a base of 40,000 is a quarter, or 25 per cent, while 10,000 on a base of 50,000 is a fifth, or 20 per cent. The two are not equal and the gap between them is a full five percentage points. The lesson generalises well beyond this question: equal absolute increases on a growing base always give a FALLING percentage rate, which is why a country whose output grows by the same amount every year is decelerating, not growing steadily.
• (c)2000 – 2010 and 1990 – 2000 — This option pairs 2000 – 2010 with 1990 – 2000, and those two are the widest apart of any pair on the list. Between 1990 and 2000 the population rose from 60,000 to 72,000, an increase of 12,000 on a base of 60,000, which is exactly one fifth or 20 per cent. Between 2000 and 2010 it rose from 72,000 to 80,000, an increase of only 8,000 on a base of 72,000, which is one ninth or about 11 per cent — the slowest decade in the table and barely half the rate of the decade before it. The option may look tempting to a candidate who has noticed that the last two figures in the column are close together and assumes the growth has settled to a steady rate; in fact the reverse has happened, since both the increase and the rate fall in the final decade. Note the printed ordering of this option, which names the later interval first; the paper sets it that way and it does not change what is being compared.
• (d)1980 – 90 and 2000 – 2010 — This option pairs the second decade with the fourth, and they are the two ends of the range rather than a match. 1980 – 90 does rise by 20 per cent, from 50,000 to 60,000, so half of the option is correct — which is precisely what makes it dangerous, because a candidate who verifies one half and assumes the other has been chosen for a reason will accept it. The other half fails badly: 2000 – 2010 takes 72,000 to 80,000, an increase of 8,000 on a base of 72,000, which is about 11 per cent, not 20. To match 20 per cent the population would have had to reach 86,400 by 2010. The habit that defends against this option is to compute ALL FOUR decadal rates before looking at the list at all. There are only four of them, they take a few seconds each with fractions rather than long division, and once they are written down every option can be checked or rejected on sight.

THE IDEA BEING TESTED IS THE DIFFERENCE BETWEEN ABSOLUTE CHANGE AND PROPORTIONAL CHANGE, which is the foundation of every growth-rate calculation in economics, demography and public finance.

An absolute change is a subtraction: this year's figure minus last year's. A proportional change is a division: the increase divided by the base, usually expressed as a percentage. The two answer different questions. The absolute change tells you how many more people, rupees or tonnes there are; the proportional change tells you how fast the quantity is growing relative to its own size. A statement that ignores which one is meant is usually misleading, and examination questions exploit that gap constantly.

THE ARITHMETIC-VERSUS-GEOMETRIC DISTINCTION follows directly. A quantity that grows by a CONSTANT AMOUNT each period forms an arithmetic progression, and its percentage growth rate FALLS every period because the base keeps rising while the numerator stays fixed. A quantity that grows at a CONSTANT RATE forms a geometric progression, and its absolute increase RISES every period. In this table the first two steps add the same amount, so the rate falls from 25 per cent to 20 per cent; the middle two steps hold the rate at 20 per cent, so the amount rises from 10,000 to 12,000.

THE TECHNIQUE that makes such questions quick is to work in ratios and simple fractions rather than percentages. 50/40 is 5/4, so a quarter more. 60/50 is 6/5, a fifth more. 72/60 is 6/5 again. 80/72 is 10/9, a ninth more. Reading a column of figures as a chain of small fractions is faster than four percentage calculations, less error-prone, and it makes matches visible at a glance.

TWO CAUTIONS worth carrying into any data-interpretation item. First, a percentage rise is always ON THE EARLIER figure; reversing the base turns a 25 per cent rise into a 20 per cent fall, which is why a rise and the fall that undoes it are never the same percentage. Second, growth rates over intervals of different length are not comparable without annualising them; here every interval is a decade, so they are directly comparable, but a table mixing five-year and ten-year gaps would need care.

Key facts

• The decadal figures are 40,000 in 1970, 50,000 in 1980, 60,000 in 1990, 72,000 in 2000 and 80,000 in 2010, printed as a ruled two-column table inside the stem.
• The four decadal percentage rises are 25 per cent for 1970 – 80, 20 per cent for 1980 – 90, 20 per cent for 1990 – 2000 and about 11 per cent for 2000 – 2010.
• The matching pair is 1980 – 90 and 1990 – 2000, both at 20 per cent, even though their absolute increases differ — 10,000 in the first and 12,000 in the second.
• The first two decades add exactly the same number of people, 10,000 each, but on bases of 40,000 and 50,000, so their percentage rises are 25 and 20 respectively.
• A percentage rise is the increase divided by the EARLIER figure, so equal absolute increases on a growing base always produce a falling percentage rate.
• A series growing at a constant rate is a geometric progression whose absolute increases rise; a series growing by a constant amount is an arithmetic progression whose rate falls.
• Working in fractions is quicker than percentages here: 50/40 is a quarter more, 60/50 and 72/60 are each a fifth more, and 80/72 is a ninth more.
• The final decade is the slowest in the table on both measures — the smallest increase, 8,000, and the lowest rate, about 11 per cent.

Common traps

• Comparing differences instead of ratios. The first two decades add 10,000 each and are the designed trap; their rates are 25 and 20 per cent.
• Verifying only one half of an option. Three of the four options name at least one interval that really does rise by 20 per cent.
• Taking the percentage on the later figure rather than the earlier one, which changes every number in the table.
• Assuming that because the last two figures are close together the growth rate has settled; the final decade is in fact the slowest of the four.
• Stopping at the first match found. All four decadal rates take only seconds to compute and should be written down before the options are read.

• practice — not a real PYQ

  The population of a town was 25,000 in the year 2000, 30,000 in 2010 and 36,000 in 2020. What were the percentage rises in population over the two decades respectively ?

  • (a)20% and 20%
  • (b)20% and 25%
  • (c)25% and 20%
  • (d)20% and 16%

  Answer(a) 20% and 20% — the rise from 25,000 to 30,000 is 5,000 on a base of 25,000, which is one fifth, and the rise from 30,000 to 36,000 is 6,000 on a base of 30,000, which is again one fifth. The absolute increases differ while the rates are equal.

• practice — not a real PYQ

  A quantity rises from 80 to 100 in one period and from 100 to 125 in the next period. Which one of the following statements about the two percentage increases is correct ?

  • (a)The first increase is the larger of the two
  • (b)The second increase is the larger of the two
  • (c)Both increases are of 25 per cent
  • (d)Both increases are of 20 per cent

  Answer(c) Both increases are of 25 per cent — 20 on a base of 80 is one quarter and 25 on a base of 100 is also one quarter, so the unequal absolute increases represent the same proportional change and the series is growing geometrically.